The Marvel Cinematic Universe (Earth-19999) has not given us a complete backstory on the Black Widow for their universe, but in the Marvel canon universe (Earth-616) she has had the equivalent of the Super-Soldier serum (a Russian variant) given to her making her the female equivalent to Captain America.
Natasha has received the Red Room's variant of the Super-Soldier Serum. As a result, her physical and mental abilities had been enhanced slightly beyond human limits.
The Black Widow is a world class athlete, gymnast, acrobat, aerialist capable of numerous complex maneuvers and feats, expert martial artist (including karate, judo, ninjutsu, aikido, savate, various styles of kung fu, and boxing), marksman, and weapons specialist as well as having extensive espionage training. She is also an accomplished ballerina.
Her agility is greater than that of an Olympic gold medalist. She can coordinate her body with balance, flexibility, and dexterity easily.
The Black Widow has been enhanced by biotechnology that makes her body resistant to aging and disease and heals at an above human rate; as well as psychological conditioning that suppresses her memory of true events as opposed to implanted ones of the past without the aid of specially designed system suppressant drugs.
Due to the Super-Soldier Serum, the white blood cells in her body are efficient enough to fight off any microbe, foreign body and others from her body, keeping her healthy and immune to most, if not all infections, diseases and disorders.
Her bodily tissues have been augmented to superhuman development. She possesses exceptional durability, strength, endurance, and stamina.
Natasha's reaction time is similarly enhanced and functions with the peak of human efficiency and capability, bordering on superhuman level. It is possible for Natasha to dodge a bullet even at point blank range. - Wikipedia -> Black Widow -> Powers
While these powers listed can be found on Wikipedia, they are a relatively recent addition to her abilities. She has always possessed a genius intellect, emotional manipulation skills, and training and was always an excellent tactician. She has been the Avengers team leader and even SHIELD leader on one occasion.
When she was first on the scene many years ago, she was little more than a highly trained human spy who used a variety of technological gadgets to aid in her spycraft. Tales of Suspense No. 52 (April 1964). She would later trick the love struck Hawkeye to her cause a few issues later. She wore a terrible fishnet costume in her early appearances.
She would attempt to steal the specifications for the Iron Man armor which would put her at odds with Tony Stark on several occasions. She would later join the Avengers and become their 16th member. Her current costume (and every variant after that) was first seen in the 1970s.
Her costume gave her a technological version of wall crawling, a utility belt (where she carried security cracking tools, shape-charges and anything else writers could squeeze on it) and she wore bracelets which fired a variety of weaponized pellets or electrical discharges she would call her Widow's sting.
She has been one of Marvel's most popular bad girls and has been the love interest of many a hero over the decades. She has worked for the US government in a variety of capacities and has an on-again, off-again relationship with SHIELD.
Her recent acquisition of a variation of the Super Soldier serum dates back to 2005 and her most recent solo comic (Black Widow No. 6, Feb. 2005) and was more than likely designed to flesh out her backstory and keep her capable of hanging with the World's Mightiest Heroes.
